Nonparametric methods in multivariate factorial designs for large number of factor levels

We propose different multivariate nonparametric tests for factorial designs and derive their asymptotic distribution for the situation where the number of replications is limited, whereas the number of treatments goes to infinity (large a, small n case). The tests are based on separate rankings for the different variables, and they are therefore invariant under separate monotone transformations of the individual variables. There are no restrictions on the covariance structure of the multivariate observations, and the methods also work for data that have ties or are measured on an ordinal scale. We compare the proposed tests to their parametric counterparts by simulating the power functions. In the presence of outliers, the new nonparametric tests show far superior power. The multivariate nonparametric tests can be used, e.g., in screening trials in agriculture or for survey data. We illustrate the application with a survey data set.